According to Proposition REF , for each \(\varrho \in \Lambda _{P}^\mathrm {spl}\) ,
we have the following isomorphism
\(&&\mathfrak {X}_{0}^{\dagger ,\varrho }\times _{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \\\cong &&((\mathfrak {Y}_{-}^{\varrho _{-},\varrho _{0}}\times \mathfrak {A}^{\varrho _{+},\varrho _{0}})\cup _{\mathfrak {A}^{\varrho _{-},\varrho _{0}}\times \mathcal {D}\times \mathfrak {A}^{\varrho _{+},\varrho _{0}}}(\mathfrak {A}^{\varrho _{-},\varrho _{0}}\times (\mathfrak {Y}_{+}^0)^{\varrho _{+},\varrho _{0}}))\times _{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \\\cong &&((\mathfrak {Y}_{-}^{\varrho _{-},\varrho _{0}}\times \mathfrak {A}^{\varrho _{+},\varrho _{0}})\times _{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho )\cup _{\mathcal {D}\times \mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho }((\mathfrak {A}^{\varrho _{-},\varrho _{0}}\times (\mathfrak {Y}_{+}^0)^{\varrho _{+},\varrho _{0}})\times _{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho ).\) 
Let \(\mathrm {pr}_{\pm }: \mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}\times \mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}\rightarrow \mathfrak {PT}_{\mathfrak {Y}_{\pm }/\mathfrak {A}}^{\varrho _{\pm },\varrho _{0}}\)  be the projections. Let \(\mathrm {ev}_{\varrho }: \mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \rightarrow \mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}\)  be the evaluation morphism and \(\iota _{\mathcal {D}}:\mathcal {D}\times \mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \hookrightarrow \mathfrak {X}_{0}^{\dagger ,\varrho }\times _{\mathfrak {C}_{0}^{\dagger ,\varrho }}\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \)  be the inclusion map of the universal distinguished divisor. Let \(\bar{\mathbb {I}}_{0}\)  be the universal ideal sheaf on \(\mathcal {D}\times \mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}\)  of the Hilbert scheme \(\mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}\) . Then on each fiber, \((\mathrm {id}_{\mathcal {D}}\times \mathrm {ev}_{\varrho })^*\bar{\mathbb {I}}_{0}\)  is quasi-isomorphic to the restriction of the universal complex \(\bar{\mathbb {I}}^\bullet _{\varrho }\)  to \(\mathcal {D}\times \mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \)  due to the admissibility.
Combined with the above isomorphism, the diagram (REF ) and Proposition REF -(iii), we have the following short exact sequence
\(0\rightarrow \bar{\mathbb {I}}^\bullet _{\varrho }\rightarrow \Delta ^*(\mathrm {pr}_{-}^*\bar{\mathbb {I}}^\bullet _{-}\oplus \mathrm {pr}_{+}^*\bar{\mathbb {I}}^\bullet _{+})\rightarrow \iota _{\mathcal {D}*}(\mathrm {id}_{\mathcal {D}}\times \mathrm {ev}_{\varrho })^*\bar{\mathbb {I}}_{0}\rightarrow 0.\) 
where \(\mathrm {pr}_{\pm }\)  and \(\Delta \)  denote the base changes to the corresponding universal spaces.
Let \(\pi _{\mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}}: \mathcal {D}\times \mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}\rightarrow \mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}\)  be the projection.
Applying \(R\pi _{\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho *}R\mathcal {H}om(\bar{\mathbb {I}}^\bullet _{\varrho }, -)^\vee \)  to the above short exact sequence, we have the following diagram of distinguished triangles
\({2pc}{g^*R\pi _{\mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}*}R\mathcal {H}om(\bar{\mathbb {I}}_{0},\bar{\mathbb {I}}_{0})_{0}^\vee [r] [d]&\bigoplus R\pi _{\mathfrak {PT}_{\mathfrak {Y}_{\pm }/\mathfrak {A}}^{\varrho _{\pm },\varrho _{0}}*}R\mathcal {H}om(\bar{\mathbb {I}}^\bullet _{\pm },\bar{\mathbb {I}}^\bullet _{\pm })_{0}^\vee [d] [r] & R\pi _{\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho *}R\mathcal {H}om(\bar{\mathbb {I}}^\bullet _{\varrho },\bar{\mathbb {I}}^\bullet _{\varrho })_{0}^\vee [d] \\\mathbb {L}_{\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho /\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}\times \mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}}[r] & \bigoplus \mathbb {L}_{\mathfrak {PT}_{\mathfrak {Y}_{\pm }/\mathfrak {A}}^{\varrho _{\pm },\varrho _{0}}/\mathfrak {A}^{\varrho _{\pm },\varrho _{0}}}[1] [r] & \mathbb {L}_{\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho /\mathfrak {C}_{0}^{\dagger ,\varrho }}[1] &}\) 
where the second row is the distinguished triangle of truncated cotangent complexes for \(\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \rightarrow \mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}\times \mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}\)  relative to \(\mathfrak {C}_{0}^{\dagger ,\varrho }\)  and the vertical arrows are induced by perfect obstruction theories. As shown in [[1]}, (6.13)] or [[2]}, Theorem 7.11], we have
\(R\pi _{\mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}*}R\mathcal {H}om(\bar{\mathbb {I}}_{0},\bar{\mathbb {I}}_{0})_{0}^\vee \cong \mathbb {L}_{\mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}/\mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}\times \mathrm {Hilb}_{\mathcal {D}}^{\varrho _{0}}}.\) 
Then combined with the Cartesian diagram (REF ), using Serre duality for the above distinguished triangles, two perfect relative obstruction theories for \(\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \)  and
\(\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}\times \mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}\)  are compatible over the diagonal morphism \(\Delta \)  in the sense of [[3]}, Section 7]. By [[3]}, Proposition 7.5], we have
\([\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho ]^\mathrm {vir}=\Delta ^!([\mathfrak {PT}_{\mathfrak {Y}_{-}/\mathfrak {A}}^{\varrho _{-},\varrho _{0}}]^\mathrm {vir}\times [\mathfrak {PT}_{\mathfrak {Y}_{+}/\mathfrak {A}}^{\varrho _{+},\varrho _{0}}]^\mathrm {vir}).\) 
Now, the proof of the second identity is completed by Proposition REF .
And the first identity is obtained by [[3]}, Proposition 7.2].
